Consider the following integrals. Ia = e-x2dx I = e-x2dx Explain why Ia and I do

Question

Consider the following integrals. Ia = e-x2dx I = e-x2dx Explain why Ia and I do not depend on x. Write another version of Ia with y being the variable of integration instead of x. Write as the product of two integrals: one in terms of x and one in terms of y. Commute the orders of products and integrals until you have written as an iterated integral. Let Ra denote the region of integration for the iterated integral in part (c). Draw a sketch of Ra. Let Ca and Ca 2 denote the circles of radius a and a 2. Show that Ca and Ca 2 inscribe and circumscribe Ra. Include a sketch. Let Ja denote the iterated integral from part (c) where the region of integration is changed from Ra to Ca.Evaluate Ja. Compute . Try to think about why J = I2. Assuming J = I2 (it is a fact, but you don't need to prove it), solve for I, the value of the original integral. The answer is I = pi, and this is an important result in statistics.

Explanation / Answer

a) because they are definate egion integral through some range and the result is a constant

b)put y=x^2

Ia becomes integral (e^-y)/sqrt(y)    from y= 0 to a^2

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